Compressive sensing lattice dynamics. I. General formalism Fei Zhou(周 非)∗ Physical and Life Sciences Directorate,
arXiv:1805.08904v1 [physics.comp-ph] 22 May 2018
Lawrence Livermore National Laboratory, Livermore, California 94550, USA Weston Nielson Department of Materials Science and Engineering, University of California, Los Angeles, California 90095-1595, USA Yi Xia Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439, USA Vidvuds Ozoli¸nˇs† Department of Applied Physics, Yale University, New Haven, CT 06520, USA and Yale Energy Sciences Institute, Yale University, West Haven, CT 06516, USA (Dated: May 24, 2018)
Abstract Ab initio calculations have been successful in evaluation of lattice dynamical properties of solids with the (quasi-)harmonic approximation, i.e. assuming non-interacting phonons with infinite lifetimes. However, it remains difficult to account for anharmonicity for all but the simplest structures. We detail a systematic information theory based approach to deriving ab initio force constants: compressive sensing lattice dynamics (CSLD). The non-negligible terms necessary to reproduce the interatomic forces are selected by minimizing the `1 norm (sum of absolute values) of the scaled force constants. The part I of this series mainly focuses on the theoretical and computational details of our approach, along with select examples.
1
I.
INTRODUCTION
The dynamics of atomic vibrations plays a central role in the structural, thermodynamic and transport properties of crystalline solids at finite temperature. The quantum theory of lattice dynamics (LD) can be traced back to the pioneering Einstein model1 and Debye model2 for the specific heat of solids at low temperature. The modern lattice dynamical model3,4 forms the theoretical basis of our quantitative understanding of lattice vibrations and their relation to macroscopic properties. With the advent of efficient density-functional theory (DFT) based methods for calculating the potential energy surface (PES) of solids, the LD model provides a bridge between atomistic quantum-mechanical calculations at zero temperature and macroscopic materials properties at finite temperature.5–8 Phonons are quasiparticles representing collective vibrations on a crystal lattice. Noninteracting phonons arise from the second order (harmonic) Taylor expansion of the PES and can be readily calculated with first-principles methods.5,9,10 Interactions beyond the harmonic approximation, e.g. phonon-phonon and electron-phonon couplings, give rise to many extremely important physical phenomena related to finite phonon lifetimes and phonon frequency shifts, such as phonon scattering, lattice thermal conductivity, phase transformations, and superconductivity. However, first principles treatment of lattice anharmonic effects is presently less ubiquitous, since a practical and systematic approach to anharmonicity has proven more challenging.11–20 In principle, the “2n + 1” theorem21 of density functional perturbation theory (DFPT)22,23 is generally applicable and gives the higher-order anharmonicity. However, such computations are cumbersome and require specialized codes that are not widely available for n > 1.24 Alternatively, the anharmonic terms may also in principle be calculated by finite-difference, but the task is prohibitively expensive due to the combinatorial explosion in the number of parameters with increasing order and interaction distance.9,25 Faced with the challenge to directly compute high-order anharmonic terms, various alternatives to directly Taylor expanding the PES have been developed. The self-consistent phonon (SCPH) approximation describes effective phonon spectra at finite temperature.26 A thermally averaged effective harmonic Hamiltonian can constructed by first-principles molecular dynamics (MD) simulation,27 or by an iterative procedure in the self-consistent ab initio lattice dynamics (SCAILD) method.28,29 These methods were applied to study 2
renormalized phonon spectra of mechanically unstable metals (bcc Li, Ti, Zr, Hf, Sc and Y) at high temperature. SCPH has been extended to calculate third-order anharmonicity in Si and FeSi.30 Errea and co-workers developed a non-perturbative, stochastic harmonic approximation to deal with strongly anharmonic systems31 by renormalizing even-order potential terms into effective harmonic interactions with a tempearture-dependent Boltzmann distribution as the weight. Free-energy of mechanically unstable high-temperature phases of transition metals have been recently calculated in Ti, Zr and Hf,32 as well as in W.33 Notably, several recent works exploited crystal symmetry to represent the PES in carefully designed functional forms. Wojdel et al expanded the PES in terms of displacement differences between pairs of atoms, and explicitly considered the coupling between strain and local displacements.34 High-temperature free energy models of ZrH2 was proposed by Thomas and Van der Ven based on symmetry-adapted cluster expansion of lattice deformation.35 Very recently, Kadkhodaei and co-workers proposed the piecewise polynomial potential partitioning (P4) method to calculate the free energy of mechanically unstable bcc Ti.36 Last but not least, methods that take advantage of advanced machine-learning techniques to construct interatomic potentials have shown great promises.37,38 Nevertheless, the original Taylor expansion based LD model remains extremely attractive. Its simple mathematical form ensures broad applicability for both different crystal types and materials properties. Our goal is to remove the main practical obstacle to its wide adoption beyond the (quasi-)harmonic level: the numerical difficulty to calculate the large number of unknown force constants. The compressive sensing lattice dynamics (CSLD) method, introduced by us in a brief letter,39 can handle compounds with large, complex unit cells and strong anharmonicity up to sixth order, including materials with harmonically unstable phonon modes. We utilize compressive sensing (CS), a technique developed in the field of information science for recovering sparse solutions from incomplete data,40 to determine which anharmonic terms are important and find their values simultaneously.41,42 The method was used in ab initio studies of δ-Pu,43 SrI2 ,44 CsPbCl(Br)3 45 and thermoelectric materials.46–48 An independent implementation by Tadano and Tsuneyuki extracted anharmonic force constants and studied temperature-dependent phonon spectra in SrTiO3 .49 The CS technique was recently used to fit interatomic potentials.50 We will provide details on CSLD in this paper, organized as follows. We lay out the general theory of lattice dynamics in a form suitable for solution by a linear equation in 3
Section II, and the CS technique in Section III. A few examples are presented in section IV. Finally we give concluding remarks in Section V. Part II of this series applies CSLD to efficient phonon calculations.
II.
THEORY: LATTICE DYNAMICS
The basic theory of lattice dynamics is well known, and a comprehensive discussion can be found in classical textbooks, e.g. Ref. 51), Here we outline the theory in an abstract, cluster-based form that is completely general in terms of expansion orders and crystal lattice symmetry, easy to keep track of, systematically improvable, and suitable to be solved numerically. This helps us manage the staggering numerical complexity and account for high order anharmonicity. More importantly, the formulation makes it easy to extend the LD model to encompass additional degrees of freedom in complicated systems.
A.
General formalism
We start with a Taylor expansion of the Born-Oppenheimer potential energy E of a crystalline solid in atomic displacements, E = E0 + Φa ua +
Φabc Φab ua ub + ua ub uc + · · · , 2 3!
(1)
where E0 is the potential energy of a reference structure without displacement, a ≡ a, i 0 is a composite index for atom a and Cartesian direction i (= 1–3), ua = ra,i − ra,i is the
displacement of atomic position ra relative to reference r0a . The second-order expansion coefficients Φab ≡ Φij (ab) = ∂ 2 E/∂ua ∂ub determine the phonon dispersion in the harmonic approximation, and Φabc ≡ Φijk (abc) = ∂ 3 E/∂ua ∂ub ∂uc is the third-order anharmonic force constant tensor (FCT). In general, an order-n FCT is defined as Φi1 ...in (a1 . . . an ) = ∂ n E/∂ua1 . . . ∂uan .
(2)
The linear term Φa is absent when the reference structure represents mechanical equilibrium. The Einstein summation convention over repeated indices is implied. Systematic calculation or fitting of the higher-order anharmonic terms is challenging due to a combinatorial explosion in the number of tensors Φ(a1 · · · an ) with increasing order n and maximum distance between the sites {a1 , . . . , an }, as well as the number of elements 4
3n in an order-n tensor. To reduce the complexity and truncate Eq. (1) to a manageable form, one may rely on physical intuition, e.g. that the largest anharmonic terms correspond to neighboring atoms with direct chemical bonds and hence are short-ranged, while long range interactions vary slowly and can be accurately described using harmonic FCTs. Once the long-range Coulombic force constants have been accounted for (details in Part II), the remaining interactions are expected to be short-ranged, i.e., they decay faster than the 3rd power of the interatomic distance.52 However, such knowledge is generally not a priori obvious in a complex system and can only be gained on a case-by-case basis through timeconsuming cycles of model construction and cross-validation. As a result, anharmonic FCTs have been calculated only for relatively simple crystals and weak anharmonicity.14,15,30 Eq. (1) can be written in a more convenient multi-index notation (details in Appendix A): E=
1 ΦI (α)uαI , α!
(3)
where α is a cluster comprised of n atoms {a1 . . . an } and I ≡ {i1 . . . in } are the correQ sponding Cartesian indices. The FCT ΦI (α) and displacement polynomial uαI ≡ k uak ik are now referenced in this compact notation. To avoid double counting, here the order in which to reference α has to be unambiguous, e.g. pre-determined by natural ordering of indices without loss of generality. We call α a proper cluster if it contains no duplicate atoms (as used in the cluster expansion model53 ), or improper otherwise, e.g. {a, a}.
B.
Independent FCT parameters
As the Taylor expansion coefficients of the crystal potential energy, force constants have to satisfy some physical constraints, namely derivative commutativity, space group symmetry, and translational and rotational invariance.51 The number of independent FCT matrix elements are reduced by these constraints, especially in solids with high symmetry.
1.
Commutativity as partial derivatives
According to Schwarz’s theorem, the FCTs as partial derivatives defined by Eq. (2) are commutative with respect to the order of taking the partial derivatives if the potential energy surface is smooth enough. We don’t have to worry about this constraint for a proper cluster 5
α, since the order of distinctive sites is pre-determined. However, if α is improper, there exists some non-trivial one-to-one indexing function π, i.e. π(1), . . . π(n) is a permutation of 1–n, that maps α to itself: π(α) = α. We have Φπ(I) (α) = ΦI (α) ∀ π(α) = α.
(4)
For instance, it is well known that an improper pair FCT satisfies Φxy (a, a) = Φyx (a, a), i.e. a symmetric matrix. This has significant impact on the long-range force constants, as discussed in Part II.
2.
Space group symmetry
In a crystalline solid, the potential energy is invariant under the space group S. As a consequence, FCTs of cluster α and its mapping sˆα under a symmetry operator sˆ are linearly related by a 3n × 3n matrix Γ: ΦI (ˆ sα) = ΓIJ (ˆ s)ΦJ (α), ΦI (α) = ΓIJ (ˆ s−1 )ΦJ (ˆ sα).
(5) (6)
To see this, first consider a proper cluster with a certain pre-determined ordering, α = {a1 . . . an }. Operation sˆ consists of a linear transformation by a 3 × 3 matrix γ, followed by a translation τ : r0 ≡ sˆr = γ · r + τ . It maps α one-to-one into {ˆ sa1 . . . sˆan }, which is in general not ordered, but rather a permutation of the ordered α0 ≡ sˆα = {a01 , . . . , a0n }, which can be referenced from the former by a0j = sˆaπ(j) , where π is an indexing function. For clarity, here we simplify the labels: uj ≡ uaj . After sˆ, the displacement at the site j of α0 is u0j = γ · uπ(j) . The potential energy of the original cluster is Φ(α) · (u1 ⊗ u2 ⊗ . . . ⊗ un ) ≡ Φi1 ...in (α)u1,i1 u2,i2 . . . un,in . 6
After transformation it becomes Φ(α0 ) · (u01 ⊗ u02 ⊗ . . . ⊗ u0n ) = Φj1 ...jn (α0 )γij11 uπ(1),i1 . . . γijnn uπ(n),in n 1 = Φj1 ...jn (α0 )γijπ(1) uπ(1),iπ(1) . . . γijπ(n) uπ(n),iπ(n) n 1 = Φj1 ...jn (α0 )γijπ(1) · · · γijπ(n) u1,i1 · · · un,in ,
where we changed summation indices. Comparing the above expressions, the following must hold: n 1 Φi1 ...in (α) = γijπ(1) · · · γijπ(n) Φj1 ...jn (ˆ sα).
(7)
n 1 ΓIJ (ˆ s−1 ) = γijπ(1) · · · γijπ(n)
(8)
or
Since the matrix γ of symmetry operation sˆ is orthogonal, i
ΓIJ (ˆ s) = γ¯ijπ1−1 (1) · · · γ¯ijπn−1 (n) = γj1π
−1 (1)
i
· · · γj1π
−1 (1)
1 n = γjiπ(1) · · · γjiπ(n) = ΓJI (ˆ s−1 ).
(9)
Γ is therefore also orthogonal. Additionally, if cluster α is improper, π is not unique but rather allows arbitrary permutations of indices belonging to any repeated site. Taking into account the commutativity relation in Eq. (4), the above derivation obviously holds for improper clusters. In particular, it is not surprising that FCTs remain identical under a translation by lattice vectors bk (γ = 1, Γ = 1) ΦI (α + Nk bk ) ≡ ΦI (α + R) = ΦI (α).
(10)
Now we divide the clusters of the lattice into orbits under space group S. The orbit of cluster α is defined as the set of clusters into which α can be transformed by S: Sα ≡ {ˆ sα|ˆ s ∈ S} Given the linear relationship in Eq. (5) between symmetrically equivalent clusters, determination the FCT’s of an entire orbit Sα is reduced to (1) finding the FCT of a representative or symmetrically distinctive cluster α, e.g. by selecting one with as as many atoms in the 7
primitive cell as possible, and (2) calculating the FCT of other clusters in Sα by applying Eqs. (5) and (9). Hence it suffices to focus on the FCT of all symmetrically distinctive representative clusters. The latter, essentially the set of all orbits under S and hereafter used indistinguishably, will be called the orbit space A/S where A denotes the set of all possible clusters. Now Eq. (3) can be rewritten as X 1 X 0 ΦI (α0 = sˆα)uαI E= α! α0 ∈Sα α∈A/S
=
X 1 X ΓIJ (ˆ s)ΦJ (α)usIˆα , α! sˆα∈Sα
(11)
α∈A/S
where the first summation is over all representative α ∈ A/S and the second one over clusters in the orbit Sα. Given a representative cluster α, the FCT may be further simplified. We define the isotropy group Sα as the subset of S that maps α to itself: Sα ≡ {ˆ s ∈ S|ˆ sα = α} For example, Sα of an improper α = {a, . . . , a} is identical to the point group of site a. According to Eq. (5), Φ(α) satisfies ΦI (α) = ΓIJ (ˆ s)ΦJ (α), ∀ˆ s ∈ Sα .
(12)
As a simple example, consider pair interactions in a periodic crystal with one atom per unit cell. Each atom is at the center of inversion, and hence each pair is transformed onto itself upon inversion followed by a translation with τ = −R, where R is the lattice vector pointing from the origin to the second vertex of the pair. Since such an operation permutes the vertices of the pair, application of Eq. (7) results in the well-known symmetry condition for pair interactions in a monoatomic crystal: Φpair = ΦTpair . This also holds in multicomponent crystals for pair interactions between equivalent atoms which are at the center of inversion and separated by a lattice vector. Consider for another example the FCT Φ(a, . . . , a) in diamond cubic silicon with point group ¯43m and rock-salt NaCl with point group m¯3m. The non-zero elements are shown in Table I. As expected, harmonic force constants Φ(aa) are constrained by symmetry to be Φxx = Φyy = Φzz , i.e. isotropic. The anharmonic FCTs are more complicated. In contrast to Si, inversion symmetry in rock-salt eliminates the odd order FCT. 8
Si 1: 0
2: 1
3: 1
None
Φxx = Φyy = Φzz or xx =
xyz = xzy = · · · = zyx
yy = zz 4: 2
5: 1
6: 3
xxyy = xyxy = xyyx = · · · =
xxxyz = xxxzy = xxyxz =
xxxxxx = yyyyyy = zzzzzz;
zzyy; xxxx = yyyy = zzzz
· · · = zzzyx
xxyyzz = · · · = zzyyxx; xxxxyy = · · · = zzzzyy NaCl
1: 0
2: 1
3: 0
None
Same as Si
None
4: 2
5: 0
6: 3
Same
None
Same
TABLE I. Non-zero elements of n-th order Φ(a . . . a) in Si and NaCl after symmetrization by isotropy group. The number of degrees of freedom left is shown after n. The results for Na and Cl are identical since they have the same point group symmetry. Note here Φxx is written as xx for brevity. 3.
Translational invariance
According to the famous Noether’s theorem, translational invariance of the Hamiltonian is essential for the momentum conservation law. Irrespective of the lattice type, the invariance of the total energy upon an arbitrary, uniform translation of the crystal leads to the acoustic sum rule (ASR) for pair force constants: Φ({aa}) = −
X
Φ({ab}).
(13)
b6=a
which states that the FCT of the improper pair cluster on atom a can be obtained by summing up the FCT of all proper pairs {a, b}. The ASR can be generalized to any order as: X
ΦI ({a, b, c, · · · }) = 0
a
9
for arbitrary lattice sites b, c, · · · and cartesian indices I. Similar to Eq. (11), the above summation can be rewritten by grouping clusters into orbits X X ΦI (ˆ sα = {a, b · · · }) = ΓIJ (ˆ s)ΦJ (α) = 0, a
(14)
a
where any cluster {a, b · · · } is identified as related to representative cluster α ∈ A/S by operator sˆ. This constitutes yet another set of linear constraints on the FCTs. Analogous to the above discussions, conservation of angular momentum requires rotational invariance of the total Hamiltonian, which can be expressed as linear constraints. However, rotational invariance involves force constants of different order in the same equation and is more complicated. In this work we do not impose rotational invariance constraints explicitly.
4.
Determination of independent parameters
As discussed previously, invariance with lattice vector translation in Eq. (11) allows us to focus on the FCT of representative clusters α ∈ A/S. We denote by ΦS the one-dimensional combined list of all NΦ such FCT elements. The number of truly independent parameters can be further reduced by the requirement for symmetric cartesian indices for repeating vertices of improper clusters in Eq. (4), the space group symmetry constraints for the isotropy group Sα in Eq. (12), as well as the translational invariance constraints in Eq. (14). All three equations can be expressed in a linear equation for ΦS : BΦS = 0,
(15)
where the B matrix contains the above mentioned (possibly redundant) constraints. For example, Eq. ( 12) can be rewritten as [Γ(ˆ s) − 1] Φ(α) = 0. Note that Eqs. (4, 12) symmetrizes a single FCT while Eq. (14), the translational invariance, places constraints on different tensors of the same order. The basis vectors of the null-space of matrix B in Eq. (15) can be used to identify independent FCT parameters. Depending on the null-space construction method, the choice of independent parameters may not be unique. In this work we employ an iterative rowreduction algorithm with the three sets of constraints applied in the same order as shown in 10
the previous paragraph. If the null-space dimension or nullity of B is Nφ , we are left with Nφ independent parameters φ with which to express the original NΦ variables: ΦS = Cφ,
(16)
where C is a NΦ ×Nφ matrix. More details on this procedure can be found in Appendix B. It allows us to impose the physical constraints exactly, without having to check the numerical accuracy of e.g. the ASR. For example, consider a minimal anharmonic lattice dynamical model of silicon. It consists of nearest neighbor interactions with two symmetrically distinct pairs {aa} and {ab} and two anharmonic triplets {aaa} and {aab}, where the lattice sites are a = (000) and b = ( 14 14 14 ). There are only two independent harmonic parameters and three anharmonic ones: 1. Φ(aa) = −4φ1 1 0 2. Φ(ab) = φ2 1 1
and Φ(ab) = φ1 1; 1 1 0 1 ; 1 0
3. Φijk (aaa) = −4φ3 |ijk |, Φijk (aab) = φ3 |ijk |; 4. Φijk (aab) = φ4 (1 − |ijk | − δijk ); 5. Φijk (aab) = φ5 δijk , where ijk is the Levi-Civita function and φk ’s are the final independent parameters we seek. Obviously Φ(aa) and Φ(aaa) are consistent with Table I.
C.
Linear problem for force constants
In order to calculate the FCTs, we take advantage of the force-displacement relationship. The force Fa on atom a in direction i can be obtained from taking the derivative of Eq. (11): Fa = −∂E/∂ua X 1 X =− ΓIJ (ˆ s)ΦJ (α)∂a0 usIˆα . α! sˆα∈Sα α∈A/S
11
(17)
The forces on the left hand side can be obtained from first-principles calculations according to the Hellman-Feynman theorem using any general-purpose DFT code for a set of atomic configurations in a supercell, similar to the direct method for harmonic force constants.54 One may extract 3Na − 3 force components in a supercell of Na atoms55 , leaving us with the desired linear problem F = A0 ΦS between force components and FCT parameters. The so-called sensing (or correlation) matrix A0 of dimension NF × NΦ is calculated from atomic displacements according to A0 (a, αI) = −
1 X ΓJI (ˆ s)∂a usJˆα . α! sˆα∈Sα
(18)
Considering the independent parameters from Eq. (16), the final linear problem to solve is F = A0 Cφ ≡ Aφ,
(19)
and the sensing matrix A for independent variables φ is NF × Nφ dimensional. Once the desired φ is obtained, any FCT can be found using Eqs. (5) and (16). Alternatively, the linear equation to fit total energy is, according to Eq. (11), E = AE Cφ, 1 X AE ({u}, αI) = ΓIJ (ˆ s)usIˆα . α! sˆα∈Sα
(20)
This was done for CSLD phonon calculations in δ-Pu43 when accurate forces were not available.
D.
Pairwise potential between bonded atoms
The anharmonic force constants can be directly used to calculate phonon lifetimes and lattice thermal conductivity in weakly anharmonic materials with perturbation theory. In principle one may directly employ LD with high order anharmonicity to study macroscopic materials properties with multi-scale modeling techniques such as classical Monte Carlo (MC) or Molecular Dynamics (MD). However, there are certain circumstances where one needs to go beyond the conventional LD model. First, the Taylor expansion is accurate only within a limited range of displacement. This is illustrated with the example of one dimensional Taylor expansion of covalent bonding, represented by a Morse potential in Fig. 1. At elevated temperature and large atomic deformations, there is substantial deviation in the 12
0.0 -0.2
2nd order expansion
4
6 8
-0.4 V
Morse
-0.6 -0.8 -1.0 0.5
1.0
1.5 rr0
2.0
2.5
FIG. 1. Taylor expansion of the Morse potential.
6th and even 8th order expansions, a reflection of the inherent limitation of Taylor expansion. When the displacement is large enough with energy ∆E, the expansion may completely break down, with probability e−∆E/kB T . This probability increases quickly at high temperature, making a conventional LD intrinsically problematic for multi-scale modeling. Second, to improve the accuracy and applicable displacement range by increasing the maximum expansion order nmax is computationally demanding. We find it difficult to go beyond 6th order in practice. The next even order FCT contains 38 = 6561 elements, and the transformation matrix Γ is 38 × 38 dimensional and takes more than 300 Megabytes of memory. Since the number of clusters also explode quickly with order n, we have chosen to truncate all the Taylor expansions in this work at nmax = 6 to keep the expansion manageable. This choice is found sufficient for the relatively harmonic systems like Si and NaCl at up to 600 K. In strongly anharmonic materials such as Cu12 Sb4 S13 (tetrahedrite),39 this truncation may have a noticeable detrimental effect on the accuracy of the lattice dynamical model. In deed, our MD simulations based on the 6-th order expansion for tetrahedrite constantly crashed due to unphysical covalent bond breaking at 300 K.39 To solve this problem, we introduced pairwise force field (FF) potentials39 to augment 13
the lattice dynamical model of tetrahedrite: E = ELD + EFF = ELD +
X
Eab (rab ),
(21)
a↔b
where ELD is the normal Taylor expansion of Eq. (1), the summation goes over covalently bonded atoms a, b, and rab = |ra − rb |. Each pair potential Eab is expanded as 0 )/rc ), Eab (rab ) = l pl ((rab − rab
(22)
where pl and l are the l-th single-variable basis function and the corresponding coefficient, respectively, and r0 is the equilibrium bond length. In this work Legendre polynomials are used as basis functions. The correlation matrix AFF between force components and coefficients {} can be written for each training structure 0 AFF (a, l) = −p0l ((rab − rab )/rc )rab /(rab rc ),
(23)
and appended to the LD correlation matrix in Eq. (19) to fit the unknown parameters {φ} and {} in an expanded linear equation: F = (A, AFF )
φ
.
The main advantage of this optional step is that we gain some knowledge of high order anharmonicity by adding only a few coefficients l rather than 3n FCT elements. In MC or MD simulations, the pair potential is continuously extrapolated outside a reasonable range with a functional form Eab (r) ∼ 1/rm where m = 1 for bond stretching and m = 6 for compression. l pl ((r − r0 )/rc ), if − rc ≤ r − r0 ≤ rc Eab (r) = e> + x> /r, if r − r0 > rc e + x /r6 , if r − r0 < −rc < < There appears to be a drawback with this hybrid LD-FF approach: since the Taylor expansion is based on a complete basis set, inclusion of another set of basis functions might be counter-productive because the obtained φ and coefficients are no longer uniquely determined. In deed, the combined series of Eq. (21) is not a basis, but a over-complete frame. Fortunately, a complete basis is not a requisite condition for CS to work: tight frame is known to be compatible.56 Our results show that this combined LD-FF expansion can sustain accurate and robust MD simulation of Cu12 Sb4 S13 for very long durations. 14
E.
Lattice Molecular dynamics
A classical MD program (LMD) with Eq. (1) or optionally Eq. (21) as the interatomic potential has been developed. Multiple methods were implemented for calculating κL , including the Green-Kubo linear response formula,57,58 reverse non-equilibrium MD (RNEMD)59 and homogenous non-equlibrium MD (HNEMD) proposed by Evans.60 While all methods yielded similar results, we found after extensive testing that HNEMD was the most efficient. In HNEMD, the equations of motion are modified so that the force on atom a is given by Fa = Fa −
X
Fab (rab · Fe ) +
b
1 X Fbc (rbc · Fe ) , N b,c
(24)
where Fa is the unmodified force calculated from Eq. (17) and Fab is the force on atom a due to b. Contributions from third- and higher-order interactions to Fab were obtained by partitioning the energy evenly among all atoms in the cluster, including repeated sites. The external field Fe has the effect of driving higher energy (hotter) particles with the field and lower energy (colder) particles against the field, while a Gaussian thermostat is used to remove the heat generated by Fe . This results in a non-zero average heat flux given by Zt hJ(t)i = −βV
ds hJ(t − s) ⊗ J(0)i · Fe .
(25)
0
As Fe → 0, one recovers the linear response limit described by the Green-Kubo formula.57,58 For cubic systems the external field can be set to Fe = (0, 0, Fz ), and in the limit of t → ∞ we get the following relation: V κL = kB T 2
Z∞
− hJz (∞)i . Fz →0 T Fz
dt hJz (t)Jz (0)i = lim
(26)
0
The process then involves a series of simulations at varying external fields Fe and constant T , with a simple linear extrapolation to zero field resulting in the true κL .
F.
Perturbation theory for phonon interactions
In contrast to the classical molecular dynamics in real space, perturbation theory for phonon interactions have been well formulated in reciprocal (momentum) space.61 Based on Boltzmann transport equation (BTE) under the relaxation time approximation (RTA), 15
thermal conductivity tensor element can be obtained via summing over contributions from different phonon modes:61 X 1 f 0 (f 0 + 1)(~ωλ )2 vλi vλj τλ , kB T 2 ΩN λ λ λ
κij L =
(27)
where N is the number of included phonon modes, Ω is the volume of the primitive cell, fλ0 is the Bose-Einstein distribution function, and ωλ , vλi and τλ are frequency, group velocity component and relaxation time of phonon mode λ. The relaxation time is the reciprocal of total scattering rates, which can be calculated via Fermi’s golden rule.61 Considering intrinsic three-phonon scattering processes (λ ± λ0 → λ00 ), the single mode relaxation time can be expressed as12 + − X 1X − 1 + Γ Γ 0 00 = + λλ0 λ00 τλ0 2 λ0 λ00 λλ λ 0 00
(28)
λλ
where
Γ+ λλ0 λ00
and
Γ− λλ0 λ00
are scattering rates from absorption (+) and emission (−) processes,
which can be evaluated if harmonic phonon dispersion and third-order force constants are known.12 Γ± λλ0 λ00
0 0 f − f ~π (3) 2 λ0 λ00 × V±λλ0 λ00 = 4 f 00 + f 000 + 1 ×
δ (ωλ ± ωλ − ωλ ) , ωλ ωλ0 ωλ00 0
00
0
(3) V±λλ0 λ00
=
(29)
λ
λ
X X a,`0 b,`00 c ijk ±iq0 ·R`0
·e
e
00
−λ λa,i ±λ b,j c,k Φijk (0a, ` b, ` c) √ ma mb mc 0
00
(30)
−iq00 ·R`00
where λa,i is the i component of polarization vector of atom a, q represents the phonon wave vector of mode λ. Refined interactive scheme can be used to obtain relaxation time which takes into account the nonequilibrium states of interacting phonons.12,18 τλ = τλ0 (1 + ∆λ )
(31)
where ∆λ is the deviation from the single mode relaxation approximation. + 1 X + ∆λ = Γ 0 00 (ξ 00 τ 00 − ξλλ0 τλ0 ) N λ0 λ00 λλ λ λλ λ − 1 X1 − + Γ 0 00 (ξ 00 τ 00 + ξλλ0 τλ0 ) N 0 00 2 λλ λ λλ λ λλ
where N is the number of sampling points and ξλλ0 ≡ ωλ0 vλz 0 /ωλ vλz . 16
(32)
III.
THEORY: COMPRESSIVE SENSING
In this section we focus on the numerical solution of the linear problem F = Aφ of Eq. (19). A is an NF × Nφ matrix, where NF is the number of calculated force components, and Nφ is the total number of unknown model parameters. In practice, the latter may far exceed NF , makeing Eq. (19) underdetermined. A reasonable approach would be to choose φ so that it reproduces the training data F to a given accuracy with the smallest number of P nonzero FCT components, i.e. the so-called `0 norm kφk0 ≡ I,φI 6=0 1. Unfortunately, this is a computationally intractable problem. We have recently shown that a similar problem in alloy theory, the cluster expansion (CE) method for configurational energetics,53 can be solved efficiently and accurately using compressive sensing.41,42 CS has revolutionized information science by providing a mathematically rigorous recipe for reconstructing S-sparse models (i.e., models with S nonzero coefficients out of a large pool of possibles, N , when S N ) from only O(S) number of data points.62–64 Given training data, CS automatically picks out the relevant signals, i.e. expansion coefficients in our physics models, and determines their values in one shot. The linear problem F = Aφ in Eq. (19) is solved by minimizing the `1 norm of the coefficients, X kφk1 ≡ |φi |, i
while requiring a certain level of accuracy for reproducing the data. The `1 norm serves as a computationally feasible approximation to the `0 norm and results in a tractable convex optimization problem. Mathematically, the solution is found as φCS = arg min kφk1 + φ
µ kF − Aφk22 , 2
(33)
where the second term is the usual sum-of-squares `2 norm of the fitting error for the training data (in this case, DFT forces). The `1 term drives the model towards solutions with a small number of nonzero FCT elements, and the parameter µ is used to adjust the relative weights of the `1 and `2 terms. Higher values of µ will produce a least-squares like fitting at the expense of denser FCTs that are prone to over-fitting, while small µ will produce very sparse under-fitted FCTs, simultaneously degrading the quality of the fit. The main advantages of CLSD over other methods for model building are that it does not require prior physical intuition to pick out potentially relevant FCTs and the fitting procedure is very robust with respect to both random and systematic noise.64 17
The optimal value of µ that produces a model with the highest predictive accuracy lies between the aforementioned extremes and can be determined by monitoring the predictive error for a leave-out subset of the training data which is not used in Eq. (33).41 The predictive accuracy of the resulting model is then validated on a third, distinct set of DFT data, which we refer to as the “prediction set”. This procedure was described in detail in Ref. 41. To solve Eq. (19) with CS, all the FCTs need to have the same unit of force. We use dimensionless displacements by substituting u → u/u0 , where u0 is conceptually a “maximum” displacement chosen to be on the order of the amplitude of thermal vibrations. An order-n FCT is then scaled by Φ → Φu0n−1 in Eq. (19).
A.
CS Solver
Development of solvers for sparse signal recovery remains a highly active research area65 . We adopted the split Bregman algorithm,66 which was previous used in CS fitting of the cluster expansion model.41 The convergence rate of the convex minimization step in the split Bregman algorithm is strongly influenced by the condition number of Q = AT A + λµI. It can be improved significantly by using a suitable preconditioner. Right preconditioner: If one computes s largest eigenvectors of AT A, an efficient preconditioner can be constructed using 1
1
Cp = VT (D + λµI)− 2 V + (λµ)− 2 NT N,
(34)
where D is an s×s diagonal matrix of eigenvalues, V is an s×Nφ matrix of the corresponding eigenvectors, and N is an (Nφ − s) × Nφ matrix containing the vector space complement of V. In our experience, appreciable speed-up can be achieved even if s is a fraction of the number of equations NF (typically, 1/4) due to rescaling of the few largest eigenvalues of AT A. After variable substitution φ = Cp φ0 , the problem to be solved becomes F = ACp φ0 . The main advantage of the right preconditioner is that the objective function remains the original sum-of-squares of residuals, and the expense of the preconditioning step versus the split Bregman iteration can be controlled by selecting s, the number of eigenvalues to be computed to construct the preconditioner. 18
B.
Training structures
A key ingredient of CSLD is the choice of atomic configurations for the training and prediction sets. One of the most profound results of CS is that a near-optimal signal recovery can be realized by using sensing matrices A with random entries that are independent and identically distributed (i.i.d.).40 For the discrete orthogonal basis in the CS cluster expansion,41,42 i.i.d. sensing matrices A could be obtained by enumerating all ordered structures up to a certain size and choosing those with correlations that map most closely onto quasi-random vectors on the unit sphere. Since the Taylor expansion employs non-orthogonal and unnormalized basis functions of a continuous variable, un , this strategy is difficult to adapt for CSLD. It is challenging to construct training configurations that give A with quasi-i.i.d. entries. As a result, we require larger training sets. Nevertheless, this is not a serious limitation because a large number of independent forces (3m − 3) can be extracted from each m-atom supercell configuration.
It is intuitively appealing to use snapshots from ab initio MD (AIMD) trajectories since they represent low-energy configurations. However, physically accessible low-energy configurations under thermodynamic distributions give rise to strong cross-correlations between the columns of A (i.e., high mutual coherence of the sensing matrix67 ), which decreases the efficiency of CS due to the difficulty of separating correlated contributions to F from different FCTs. To solve this conundrum, we combine the physical relevance of AIMD trajectories with the mathematical advantages of efficient compressed sampling by adding random displacements (∼ 0.1–0.2 ˚ A) to each atom in sufficiently spaced snapshots from short AIMD trajectories. This procedure was found to decrease the coherence as measured by the cross-correlations between the columns of the sensing matrix A,67 resulting in stable signal recovery.
For the relatively simple task of fitting harmonic force constants for phonon spectra, we found it sufficient to independently displace all atoms in the training supercell structure in a random direction by 0.01 ˚ A away from equilibrium. 19
C.
Fitting in steps
In practical CSLD fittings, one may wish to adopt a divide-and-conquer strategy to fit different groups of parameters in steps for two reasons. First, as a consequence of the non-orthogonal and unnormalized basis functions in the Taylor expansion and the optional pairwise functions, the numerical stability of fitting is reduced. Secondly, in complex structures with high-order anharmonicity, the number of independent parameters Nφ can easily >3000), one exceed 104 , making direct fitting in one shot very inefficient. If Nφ is large (∼ may therefore divide φ into subsets ψ 1 , ψ 2 , . . . and the sensing matrix into corresponding sub-matrices A = (A1 , A2 , . . . ), e.g. pairwise potential parameters , harmonic parameters φ(2) , third-order φ(3) , etc. The parameters ψ n are fitted sequentially, taking into account the contribution of previously obtained ones: F − (A1 , . . . , An−1 ) ψ T1 , . . . , ψ Tn−1
T
= An ψ n .
(35)
Training structures can be adapted for each set of parameters: small displacement of 0.01 ˚ A for harmonic φ(2) , gradually larger displacement from higher temperature AIMD snapshots for higher order anharmonic terms. This procedure was used for fitting Cu12 Sb4 S13 .
IV.
RESULTS AND DISCUSSIONS
All DFT calculations were performed using the Perdew-Becke-Ernzerhof (PBE) functional,68 PAW potentials,69 a cutoff energy of 600 eV, energy convergence tolerance of 10−9 eV per atom, and no symmetry constraints as implemented in the VASP code.70 Lattice parameters were fixed at experimental values. AIMD simulation was run in 1 fs steps with lowered computational accuracy (smaller cutoff and larger tolerance) and snapshots were taken at 3ps intervals and re-calculated with high accuracy. We show in Fig. 2 the results of third order fitting for cubic silicon, rock salt and aluminum. In each case two 3×3×3 fcc supercell structures, with all atoms randomly displaced by 0.03 ˚ A, were used for CSLD fitting, including all possible second and third order clusters with diameter not exceeding half the size of the cell. Here the diameter is defined as the maximum distance of pairs in the cluster, dα = maxa,b∈α dab . By far the most significant third order FCT found is the improper cluster in Si, aaa, which does not vanish due to the absence of inversion symmetry. We found that the magnitude of FCTs involving two atoms 20
△ ●
Φ (α )
��� (��/Å
�
)
(a) ��� �� �� � � ���
△
α=aaa
●
α=aab
▲
α=abc
▲
�
�
● ●
●
▲ ▲
▲ ▲
�
�
� �
▲ ▲ ▲
● ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
● ▲ ▲ ▲ ▲ ● ▲ ▲ ▲
�/�
�
�α/�� ● ●
Φ (α)
��� (��/Å
�
)
(b)
● ▲ ▲
� ����
● ●
���� ����
▲ ▲
●
● ▲
▲ ● ▲
▲ ▲ ▲ ▲ ●
▲
● ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
▲
����
� �
● ▲ ▲ △ ▲ ▲ ▲ ▲ ▲ ▲
� �
�
�/�
�α/�� �
●
Φ (α)
��� (��/Å
�
)
(c)
� ���� ���� ����
▲ ●
21▲
● ▲ ▲ ▲
● ▲ ▲ ▲ ▲
(aab, filled circles) are generally larger than proper clusters abc (filled triangles). That the improper third-order FCTs are larger than proper ones is also observed in harmonic force constants and can be attributed to the generalized ASR in Eq. (14), which relates the improper FCTs to proper (and less “improper”) ones. Φ(aab) drops in magnitude quickly as the interaction distance increases (note the log scale). In fact, the most appreciable Φ(aab) on the nearest-neighbor ab pair is about 30–60 times larger than the second nearest neighbor and beyond, suggesting that compared to the harmonic terms, the anharmonic ones are even more short-ranged. Cu(2)-S(1)
Cu(1)-S(1) 2.0 E-Emin (eV)
E-Emin (eV)
2.0 1.5 1.0 0.5
1.0 0.5
0.0
0.0 -0.5
0.0
0.5
-0.5
1.0
0.0
0.5
r-r 0 ()
r-r 0 ()
Cu(2)-S(2)
Sb-S(1)
1.0
2.0 E-Emin (eV)
2.0 E-Emin (eV)
1.5
1.5 1.0 0.5
1.5 1.0 0.5
0.0 -0.5
0.0
0.5
1.0
0
0.0 -0.6 -0.4 -0.2
0.0
0.2
0.4
0.6
0
r-r ()
r-r ()
FIG. 3. Obtained pairwise potential EFF for Cu12 Sb4 S13 .
The earth-abundant natural mineral tetrahedrite (Cu12 Sb4 S13 ) has been recently shown to be a high-performance thermoelectric.71,72 The body-centered cubic (bcc) structure with space group I ¯43m has 29 atoms in the primitive cell, a large number that complicates the computation of FCTs. For example, there are 188 distinct atomic pairs within a radius of a = 10.4 ˚ A, 116 triplets within a/2, etc. Taking into account the 3n elements of each tensor, 22
the number of unknown FCT coefficients is very large (55584 in our setting). After symmetrization, this is reduced to 3188, which still represents a formidable numerical challenge. Fig. 3 shows the obtained force-field potential EFF of bonded cation-anion pairs, including results for two sub-lattices for each of copper and sulfur. Up to 12 Legendre polynomials and a scaling length rc was used in the fitting. The final LD+FF fitting and lattice thermal conductivity results were previously reported39 and not repeated here.
Ì
Û
14
kL HW m-1 K-1 L
12
Ê ‡ Green-Kubo
Ì
Û 10 — ‡ Ì Ê — Ì 8 Û — ‡ · — Á 6 Û
Ê BTE
Ì Klein Û Håkansson · Yang Á Clark
Ê
—
Ê
‡ ÁÛ — · ÁÛ Á
4
— ‡ · — Á
Á
2 0
200
300
Ê
— · ‡ Á— Á
400 500 600 Temperature HKL
Ê
Á·
700
Ê
·
800
FIG. 4. Comparison of lattice thermal conductivities of NaCl versus temperature using CSLDderived Green-Kubo formula and PT/BTE, as well as experimental measurements from Refs. 73– 76. The dash black lines in the figure are drawn to guide the eye.
Finally, we point out the importance to include high-order interatomic interactions in order to accurately model lattice heat transport, through a comparative study of lattice thermal conductivity of NaCl using PT/BTE and Green-Kubo linear response formula, respectively. Green-Kubo molecular dynamics simulations, based on an up to fourth-order CSLD fitting, were performed between 200 and 600 K, with system sizes ranging from 512 to 4096 atoms. The lengths of the simulations are from 100 ps to 1 ns with a timestep of 23
1 fs. At each temperature, a minimum of 10 independent configurations were used. No discernible size-dependence was found in the range of supercells tested. Fig. 4 shows that the thermal transport in NaCl is nontrivial, indicating that perturbation theory always overestimates κ compared to experiment in the entire range of simulated temperatures, mainly due to underestimated phonon scattering rates. Meanwhile, Green-Kubo formula gives significantly reduced κ, achieving much better agreement with experiment above the Debye temperature (≈ 300 K for NaCl) and further confirming the importance of highorder anharmonicity (fourth-order IFCs). The tendency to underestimate κ of Green-Kubo formula at lower temperatures probably can be attributed to the lack of quantum correction in classical MD, as detailed in Ref. 77.
V.
CONCLUSION
We have described in detail both the lattice dynamical model in a cluster-based form that is convenient for keeping track of high order force constants, and the compressive sensing framework tailored towards force constants extraction from DFT calculations. For instance, CSLD can easily include 4–6 th-order anharmonicity (important for many cubic systems with double-well type potentials), which is inaccessible to DFPT and “2n+1” methods. CSLD is more general, efficient and straight-forward than the existing methods for treating anharmonicity. The model accuracy can be improved systematically by simply increasing the size of the training set. The software package CSLD will be made publicly available in the near future. Applications of CSLD for phonon calculations will be presented in Part II of the series. Beyond lattice dynamics in crystalline solids, the formalism developed in this work is being extended to encompass other degrees of freedom such as substitutional defects.
ACKNOWLEDGMENTS
The work of F.Z. was supported by the Laboratory Directed Research and Development program at Lawrence Livermore National Laboratory and the Critical Materials Institute, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office, and performed under the 24
auspices of the U.S. Department of Energy by LLNL under Contract DE-AC52-07NA27344. V.O. acknowledge support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Grant DE-FG02-07ER46433, and computational resources from the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
∗
[email protected]
†
[email protected]
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Appendix A: Multi-index notation
In the multi-index notation for N variables, an N -tuple α = (α1 , . . . , αN ) of non-negative integers is defined with the following notations for its absolute value, factorial, power, and partial derivative, respectively: |α| =
X
αa
(A.1)
αa !
(A.2)
uαa a
(A.3)
a
α! =
Y a
uα =
Y a
∂uα = ∂ |α| /
Y
∂uαa a .
(A.4)
a
The N -tuple α is equivalent to a “flattened” list (hereby called cluster) C(α) of length |α| C(α) = {a, . . . , a, a, . . . } where element a appears exactly αa times. Hereafter we refer to the N -tuple α and the corresponding |α|-element cluster C(α) indistinguishably. In a proper cluster αa ≤ 1. 28
In this work, the N -tuple is used for designating a cluster of atoms. When used in conjunction with another list of index I = {i}, we define uαI
=
|α| Y
uak ,ik
k=1
For example, given a proper pair cluster composed of sites s1 and s2 : α = (1, 1, 0, . . . ) = {s1 , s2 }, α! = 1, ΦI (α) = Φ1i1 ,2i2 , and uαI = u1i1 u2i2 . Given an improper triplet cluster of site s1 alone: α = (3, 0, . . . ) = {s1 , s1 , s1 }, α! = 6, ΦI (α) = Φ1i1 ,1i2 ,1i3 , and uαI = u1i1 u1i2 u1i3 . According to above definitions the potential expansion of Eq. (1) can be conveniently translated into Eq. (3).
Appendix B: Iterative null-space construction
Given a set of Nc linear constraints on the Nv -dimensional variable Φ such that certain linear combinations of Φ vanish: Bk Φ = 0, where each Bk is an nk × Nv matrix, k = 1 . . . Nc , the number of independent variables is reduced from Nv to the dimension of the (right) null space of the full constraint matrix B composed of all constraint matrices BT = (BT1 , . . . , BTNc ), and the independent variables may be chosen according to the basis vectors {c1 , . . . } of the null space. An iterative procedure to solve for these basis vectors in columns C = (c1 , . . . ) is: 1. Initialize C = 1Nv , tolerance δ (e.g. 10−8 ) 2. For k in 1–Nc (a) Row-reduce Bk C by Gauss-Jordan elimination into a row echelon form to find its null space basis vectors in columns, C0 . To enhance numerical stability, elements of C0 with absolute values below δ are set to 0. (b) Update C ← CC0 . 3. Return C.
29
